Download Analysis of Low-Frequency Electromagnetic Devices using Finite

Analysis of Low-Frequency Electromagnetic Devices
using Finite Elements
Ingeniería Energética y Electromagnética
Escuela de Verano de Potencia 2014
Rafael Escarela (UAM-A)
Coupling Circuit and Field Systems
August 2014
1 / 38
Overview
The nodal method and its limitations.
Modified nodal analysis (MNA): Implementation basics.
Low-frequency electromagnetic equations.
Circuit-field coupled problem.
Example: Induction machine operating at steady state.
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Coupling Circuit and Field Systems
August 2014
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Nodal Method
Algorithm
The system of circuit equations is easily constructed through
building blocks of admittance elements.
It is widely used in power network calculations.
Forcing currents injected at specified circuit nodes.
Building blocks are obtained from:
Network-independent voltage-current relations (VCR) of an
element.
The numbering of the element terminals within an arbitrary network.
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August 2014
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Nodal Method
Frequency Domain
Consider only time-harmonic computations for the sake of
simplicity.
VCR of an admittance:
y a v a = ia
If the terminals are connected to nodes p and q of an arbitrary network
of m non-trivial nodes, the branch voltage can be expressed as
va = vp − vq
The current contribution to nodes p and q is given by
ip
−1
=
i
iq
1 a
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Coupling Circuit and Field Systems
August 2014
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Nodal Method
Frequency Domain
Thus the VCR can be transformed into a building block:
ya −ya
−ya ya
vp
i
= p
vq
iq
The final system of equations can be simply written as
Yv = i
Y is known as the system admittance matrix.
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Coupling Circuit and Field Systems
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Nodal Method
Attributes and Limitations
Advantages
Construction by building blocks gives an algorithm easily
programmable.
Mutually coupled admittances are readily incorporated.
Source transformation may allow incorporation of non-ideal voltage
sources.
Current controlled current sources can also be accommodated.
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Nodal Method
Attributes and Limitations
Limitations
Non-natural elements such as ideal voltage sources, dependent
sources, transformers and auto-transformers cannot be directly
considered in the analysis.
Special post- and pre- processing are required with non-natural
elements.
Loss of information with the use of transformations (such as
Norton’s theorem). Topologically connected nodes are
reduced in number.
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Coupling Circuit and Field Systems
August 2014
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The Modified Nodal Analysis
MNA
Incorporation of VCR relations as additional equations.
Injection of the element currents to nodes.
Total elimination of the nodal method limitations with the
advantage of keeping one of its main attributes: Building Block
Construction.
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Coupling Circuit and Field Systems
August 2014
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The Modified Nodal Analysis
MNA
The MNA always gives:
Y
B
A
D
v
i
=
ie
fv
Y is constructed in the usual way.
Non-natural elements are erased from the Y
building process.
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Coupling Circuit and Field Systems
August 2014
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The Modified Nodal Analysis
MNA
D, A and B are matrices whose element entries are zeros, ones,
element resistances and dependent source gains.
Zero diagonal may exist but this does not seem to be a limitation
due to efficient sparse matrix solvers and reordering techniques.
fv is a forcing vector that contains the voltage values of
independent voltage sources and zeros for other non-natural
elements.
ie is the vector of unknown branch currents of non-natural
elements.
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Coupling Circuit and Field Systems
August 2014
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The Modified Nodal Analysis
MNA
A = B in the absence of equivalent circuit dependent sources.
The MNA technique is widely used in most modern circuit
simulation packages.
It has a strong foundation on circuit theory.
The procedure for constructing the building blocks is always the
same: addition of VCR and incorporation of the element current
into the nodal equations.
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August 2014
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MNA
Building blocks
Table : Independent Active Elements
Circuit Element
Stiffness Contribution
p
q

 02×2
1
−1
s

1 p
−1  q
−zs s
.........
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Forcing Contribution
−
0
0
es
n
−is
is
o
p
q
s
p
q
August 2014
12 / 38
MNA
Building blocks
Table : Passive Elements
Circuit Element
Stiffness Contribution
p
ya
−ya
−ya p
q
k
p
q
ya q
l






04×4
1
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−1
−N
Coupling Circuit and Field Systems
N
t

1 p
−1 
q
−N 
k
N l
t
0
August 2014
13 / 38
MNA
Building blocks
Table : Cont’d
Circuit Element
Stiffness Contribution
p
q
k

1
−(N + 1)
p
ya
 −y a

 ym
−ym

Rafael Escarela (UAM-A)
q
−ya
ya
−y m
ym
Coupling Circuit and Field Systems

p
−(N + 1)
q
03×3



a
1
N
k
ym
−y m
yb
−y b
N
0
k
a
l
−y m

p
ym 
q
−yb  k
yb l
August 2014
14 / 38
MNA
Building blocks
Table : Dependent Sources
Circuit Element
Stiffness Contribution
p
q
k
l







04×4
0
0
1
−1
p
q
k
l


p
q
1

DF  k

−DF  l
0
f
g

p
1 
q

0 k

0 l
−DG g
−1






04×4
0
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f
−1
0
1
−1
Coupling Circuit and Field Systems
August 2014
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MNA
Building blocks
Table : Cont’d
Circuit Element
Stiffness Contribution
p
q
k
l







04×4
DE
−DE
p
q
−1
k
1
l
h
−1
0
1
−1
0
0
0
0








0
1
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04×4
0
−1
1
0
Coupling Circuit and Field Systems
e

1 p
q
−1

0 k

0 l
0 e
i

0 p
0 
q
1 
k

−1  l

0 h
−GH i
August 2014
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2D Low-Frequency Electromagnetic Equations
Let a 2D domain be constituted by pure non-conducting regions
as well as by an arbitrary number of filamentary and solid
conducting sub-domains.
Neglect displacement current and free charge.
Maxwell’s equations lead to the following diffusion type equation
∇ · ν∇A = −Jf − Js = −Jf + σ(∇V +
Rafael Escarela (UAM-A)
Coupling Circuit and Field Systems
∂A
)
∂t
August 2014
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Conductors
Solid and Filamentary
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2D Low-Frequency Electromagnetic Equations
Domains:

X  ∇ · ν∇A = 0
∇ · ν∇A = −Jf
:

2D
∇ · ν∇A = −Js = σ(∇V +
in Ωnc
in Ωfil
∂A
)
in
Ωsol
∂t
Solid conductor current equation:
Z
Z
Z
σ
vs
∂
∂
is = vs
σAdΩ =
AdΩ
dΩ −
−σ
∂t Ωsol
rs
∂t Ωsol
Ωsol ds
Solid conductor voltage equation:
∂
vs = rs is + ds
∂t
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R
Ωsol
AdΩ
Ss
Coupling Circuit and Field Systems
August 2014
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2D Low-Frequency Electromagnetic Equations
Filamentary conductor voltage equation:
R
∂ Ωfil AdΩ
vf = rf if + df nf
∂t
Sf
The field system can be set up as

 ∇ · ν∇A = 0
X 
∇ · ν∇A = − nSf fif
:
R

 ∇ · ν∇A − σ ∂A + σ ∂ AdΩ = − is
2D
∂t
∂t Ss
Ss
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Coupling Circuit and Field Systems
in Ωnc
in Ωfil
in Ωsol
August 2014
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Finite-element system
Finite-element discretization gives
−K a −
X
X
da X
(Cs − Cs0 )
+
Nf if +
Ns is = 0
dt
s
s
f
K =
XZ
e
T
ν∇ζ · ∇ζ dΩ
Ωe
1
Ns =
Ss
Coupling Circuit and Field Systems
σζ T ζ dΩ
Ωe
e
Z
nf
Nf =
ζ dΩ
Sf
σSs
ds
Cs0 =
Ns NsT =
Ns NsT
ds
Rs
Rafael Escarela (UAM-A)
Cs =
XZ
Z
ζ dΩ
August 2014
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Connecting the FE model to the exterior world
Figure : MVP controlled voltage sources and their interaction with the FE
Model: (A) Filamentary and (B) Massive.
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Coupling Circuit and Field Systems
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Magnetic vector potential controlled voltage sources
da
dt
da
vs =rs is + ds NsT
dt
R
Z
∂ Ωfil AdΩ
da
da
n
nf
= f
ζ T dΩ
= NfT
∂t
Sf
Sf Ωfil
dt
dt
R
Z
1
da
da
∂ Ωsol AdΩ
=
ζ T dΩ
= NsT
∂t
Ss
Ss Ωsol
dt
dt
vf =rf if + df NfT
Frequency domain counterparts are obtained by
∂
substituting ∂t
by jω.
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Circuit-field uncoupled problem
K 0 = K + jωCeq

   
−K 0 0 0  a   fi 
 0
Y A v = i
   
0
B D
fv
ie
P
P
and fi = − Nf if − Ns is
 

 a  
−K 0 0 0 N 
 
  fi 
v
 0

Y E 0
= i
ie 
 
0
F D 0 
fv
 

ic
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Circuit-field coupled problem
The VCR of solid and filamentary conductors can be written in the
frequency domain as
vp − vq = ri + j d ωN T a
subscripts f and s have been dropped out since filamentary and solid
conductors can be treated in a unified way
This VCR can be added to as an additional equation, leading to

   
−K 0
0
0 N 
 
 fi 

a
 0
 v   i 
Y
E
0


=
 0
F D 0 
ie 
f 






 v

ic
0
j d ωN T −`T 0 r
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August 2014
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Building Block for Filamentary and Solid Conductors
The coupling is completed adding ic
and subtracting it to equation q:

−K 0
0
0
 0
Y
E

 0
F D
T
j d ωN
−`T 0
to equation p of the nodal system
   
N 
a 
fi 


 
 
 
`
i
v

=
0 

ie 
 fv 
  
 
0
ic
r
A building block can now be determined as


0


j d ωN T −`T
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
N
`

0
0 r
Coupling Circuit and Field Systems
August 2014
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Coupled system
Inclusion of all massive and solid conductors gives the final system

   
−K 0
0
0 N 

a
 
 fi 
 0
 v   i 
Y
E
L


=
 0
F
D 0
f 
ie 







 v
0
ic
j d ωN T −LT 0 R
which can be conveniently rewritten as (κ , j d1ω )


  
−K 0
0
0
N
a
f




i
  



T  v 
 0
−κY
−κE
−κL
−κi


=
 0
−κF −κD
0 
i  
−κfv 


 e
 


T
ic
0
N
−κL
0
κR
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Coupled system
The coupled field-circuit system is symmetric if E = F.
This condition is achieved with circuits that do not contain voltage
or current controlled sources.
Notice that MVP-controlled sources do not lead to asymmetry.
The coupled problem will not have zeros in the main diagonal for
systems that consider non-ideal voltage sources and inductances,
capacitances and resistances.
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Coupled system
The incorporation of building blocks can be arbitrarily performed in
any order.
This means that the building blocks of solid and filamentary conductors
are actually members of the family of non-natural circuit elements:


  
−K 0
0
N
 a   fi 
 0
−κY −κE  v = −κi
  

T
ie
−κfv
N
−κF −κD
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Induction motor coupled to circuits
Quasi 3D FE model
The test case involves the time-harmonic non-linear operation of a
squirrel cage induction motor.
It is a two-pole, 7.5 kW, 380 V, 50 Hz, three-phase star connected
motor.
Figure : 7235 nodes and 3464 second order elements (triangular and
quadrilateral).
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Circuits
Stator and rotor
Figure : Stator connections.
Figure : Rotor connections. Antiperiodic boundary conditions are considered.
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Result
Flux Plot: Rated Operating Condition.
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Result
Table : FE Calculated Currents: Filamentary
ia
8.652∠ − 27.554◦
ib
8.596∠ − 146.746◦
ic
8.729∠93.168◦
N.B. ia , ib and ic are the RMS stator phase currents. Current
values in [A].
Table : FE Calculated Currents: Solid
Bar1
456.981∠152.485◦
Bar5
451.622∠82.930◦
Bar10
381.001∠ − 3.334◦
N.B. RMS current values in [A].
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Rotating magnetic field
Show Movie
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Conclusions
The MNA scope has been broadened out with the addition of one
new non-natural element (a MVP-controlled voltage source)
The FE model has benefited from a well-posed formulation of
unknown currents in conductor regions.
A building block has been presented for systematic inclusion of
both solid and filamentary conductors of FE regions within
arbitrary topologies of circuit systems.
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Conclusions
The circuit system does not “see” differences between these
classes of conductors, it only sees controlled voltage sources.
The proposed MNA-FE coupling technique is based on sound
circuit theory known by all electrical and electronic engineers.
Concepts of of topology theory are not needed.
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August 2014
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Conclusions
The methodology can also be easily mounted on existing FE
codes due to the building block approach that is also
conventionally used in the FE method.
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Coupling Circuit and Field Systems
August 2014
37 / 38
Gracias!
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Coupling Circuit and Field Systems
August 2014
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